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RectangularCholeskyDecomposition (Commons Math 3.2 API)
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org.apache.commons.math3.linear</FONT>
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Class RectangularCholeskyDecomposition</H2>
<PRE>
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  <IMG SRC="../../../../../resources/inherit.gif" ALT="extended by "><B>org.apache.commons.math3.linear.RectangularCholeskyDecomposition</B>
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<DT><PRE>public class <B>RectangularCholeskyDecomposition</B><DT>extends <A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</A></DL>
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<P>
Calculates the rectangular Cholesky decomposition of a matrix.
 <p>The rectangular Cholesky decomposition of a real symmetric positive
 semidefinite matrix A consists of a rectangular matrix B with the same
 number of rows such that: A is almost equal to BB<sup>T</sup>, depending
 on a user-defined tolerance. In a sense, this is the square root of A.</p>
 <p>The difference with respect to the regular <A HREF="../../../../../org/apache/commons/math3/linear/CholeskyDecomposition.html" title="class in org.apache.commons.math3.linear"><CODE>CholeskyDecomposition</CODE></A>
 is that rows/columns may be permuted (hence the rectangular shape instead
 of the traditional triangular shape) and there is a threshold to ignore
 small diagonal elements. This is used for example to generate <A HREF="../../../../../org/apache/commons/math3/random/CorrelatedRandomVectorGenerator.html" title="class in org.apache.commons.math3.random"><CODE>correlated
 random n-dimensions vectors</CODE></A> in a p-dimension subspace (p < n).
 In other words, it allows generating random vectors from a covariance
 matrix that is only positive semidefinite, and not positive definite.</p>
 <p>Rectangular Cholesky decomposition is <em>not</em> suited for solving
 linear systems, so it does not provide any <A HREF="../../../../../org/apache/commons/math3/linear/DecompositionSolver.html" title="interface in org.apache.commons.math3.linear"><CODE>decomposition solver</CODE></A>.</p>
<P>

<P>
<DL>
<DT><B>Since:</B></DT>
  <DD>2.0 (changed to concrete class in 3.0)</DD>
<DT><B>Version:</B></DT>
  <DD>$Id: RectangularCholeskyDecomposition.java 1422313 2012-12-15 18:53:41Z psteitz $</DD>
<DT><B>See Also:</B><DD><a href="http://mathworld.wolfram.com/CholeskyDecomposition.html">MathWorld</a>, 
<a href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Wikipedia</a></DL>
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<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/RectangularCholeskyDecomposition.html#RectangularCholeskyDecomposition(org.apache.commons.math3.linear.RealMatrix)">RectangularCholeskyDecomposition</A></B>(<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A>&nbsp;matrix)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Decompose a symmetric positive semidefinite matrix.</TD>
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<TR BGCOLOR="white" CLASS="TableRowColor">
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/RectangularCholeskyDecomposition.html#RectangularCholeskyDecomposition(org.apache.commons.math3.linear.RealMatrix, double)">RectangularCholeskyDecomposition</A></B>(<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A>&nbsp;matrix,
                                 double&nbsp;small)</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Decompose a symmetric positive semidefinite matrix.</TD>
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<CODE>&nbsp;int</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/RectangularCholeskyDecomposition.html#getRank()">getRank</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Get the rank of the symmetric positive semidefinite matrix.</TD>
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<CODE>&nbsp;<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A></CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../org/apache/commons/math3/linear/RectangularCholeskyDecomposition.html#getRootMatrix()">getRootMatrix</A></B>()</CODE>

<BR>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Get the root of the covariance matrix.</TD>
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<A NAME="RectangularCholeskyDecomposition(org.apache.commons.math3.linear.RealMatrix)"><!-- --></A><H3>
RectangularCholeskyDecomposition</H3>
<PRE>
public <B>RectangularCholeskyDecomposition</B>(<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A>&nbsp;matrix)
                                 throws <A HREF="../../../../../org/apache/commons/math3/linear/NonPositiveDefiniteMatrixException.html" title="class in org.apache.commons.math3.linear">NonPositiveDefiniteMatrixException</A></PRE>
<DL>
<DD>Decompose a symmetric positive semidefinite matrix.
 <p>
 <b>Note:</b> this constructor follows the linpack method to detect dependent
 columns by proceeding with the Cholesky algorithm until a nonpositive diagonal
 element is encountered.
<P>
<DL>
<DT><B>Parameters:</B><DD><CODE>matrix</CODE> - Symmetric positive semidefinite matrix.
<DT><B>Throws:</B>
<DD><CODE><A HREF="../../../../../org/apache/commons/math3/linear/NonPositiveDefiniteMatrixException.html" title="class in org.apache.commons.math3.linear">NonPositiveDefiniteMatrixException</A></CODE> - if the matrix is not
 positive semidefinite.<DT><B>Since:</B></DT>
  <DD>3.1</DD>
<DT><B>See Also:</B><DD><a href="http://eprints.ma.man.ac.uk/1193/01/covered/MIMS_ep2008_56.pdf">
 Analysis of the Cholesky Decomposition of a Semi-definite Matrix</a></DL>
</DL>
<HR>

<A NAME="RectangularCholeskyDecomposition(org.apache.commons.math3.linear.RealMatrix, double)"><!-- --></A><H3>
RectangularCholeskyDecomposition</H3>
<PRE>
public <B>RectangularCholeskyDecomposition</B>(<A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A>&nbsp;matrix,
                                        double&nbsp;small)
                                 throws <A HREF="../../../../../org/apache/commons/math3/linear/NonPositiveDefiniteMatrixException.html" title="class in org.apache.commons.math3.linear">NonPositiveDefiniteMatrixException</A></PRE>
<DL>
<DD>Decompose a symmetric positive semidefinite matrix.
<P>
<DL>
<DT><B>Parameters:</B><DD><CODE>matrix</CODE> - Symmetric positive semidefinite matrix.<DD><CODE>small</CODE> - Diagonal elements threshold under which columns are
 considered to be dependent on previous ones and are discarded.
<DT><B>Throws:</B>
<DD><CODE><A HREF="../../../../../org/apache/commons/math3/linear/NonPositiveDefiniteMatrixException.html" title="class in org.apache.commons.math3.linear">NonPositiveDefiniteMatrixException</A></CODE> - if the matrix is not
 positive semidefinite.</DL>
</DL>

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<A NAME="getRootMatrix()"><!-- --></A><H3>
getRootMatrix</H3>
<PRE>
public <A HREF="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</A> <B>getRootMatrix</B>()</PRE>
<DL>
<DD>Get the root of the covariance matrix.
 The root is the rectangular matrix <code>B</code> such that
 the covariance matrix is equal to <code>B.B<sup>T</sup></code>
<P>
<DD><DL>

<DT><B>Returns:</B><DD>root of the square matrix<DT><B>See Also:</B><DD><A HREF="../../../../../org/apache/commons/math3/linear/RectangularCholeskyDecomposition.html#getRank()"><CODE>getRank()</CODE></A></DL>
</DD>
</DL>
<HR>

<A NAME="getRank()"><!-- --></A><H3>
getRank</H3>
<PRE>
public int <B>getRank</B>()</PRE>
<DL>
<DD>Get the rank of the symmetric positive semidefinite matrix.
 The r is the number of independent rows in the symmetric positive semidefinite
 matrix, it is also the number of columns of the rectangular
 matrix of the decomposition.
<P>
<DD><DL>

<DT><B>Returns:</B><DD>r of the square matrix.<DT><B>See Also:</B><DD><A HREF="../../../../../org/apache/commons/math3/linear/RectangularCholeskyDecomposition.html#getRootMatrix()"><CODE>getRootMatrix()</CODE></A></DL>
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